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Properties

Label 2-325-5.4-c5-0-82
Degree 22
Conductor 325325
Sign −0.447−0.894i-0.447 - 0.894i
Analytic cond. 52.124752.1247
Root an. cond. 7.219747.21974
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.83i·2-s − 12.7i·3-s − 2.07·4-s − 74.5·6-s − 231. i·7-s − 174. i·8-s + 80.0·9-s − 223.·11-s + 26.5i·12-s + 169i·13-s − 1.35e3·14-s − 1.08e3·16-s − 1.32e3i·17-s − 467. i·18-s + 1.48e3·19-s + ⋯
L(s)  = 1  − 1.03i·2-s − 0.818i·3-s − 0.0649·4-s − 0.845·6-s − 1.78i·7-s − 0.964i·8-s + 0.329·9-s − 0.556·11-s + 0.0531i·12-s + 0.277i·13-s − 1.84·14-s − 1.06·16-s − 1.11i·17-s − 0.339i·18-s + 0.940·19-s + ⋯

Functional equation

Λ(s)=(325s/2 ΓC(s) L(s)=((−0.447−0.894i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(325s/2 ΓC(s+5/2) L(s)=((−0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 325325    =    52⋅135^{2} \cdot 13
Sign: −0.447−0.894i-0.447 - 0.894i
Analytic conductor: 52.124752.1247
Root analytic conductor: 7.219747.21974
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ325(274,⋅)\chi_{325} (274, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 325, ( :5/2), −0.447−0.894i)(2,\ 325,\ (\ :5/2),\ -0.447 - 0.894i)

Particular Values

L(3)L(3) ≈\approx 2.1734507622.173450762
L(12)L(\frac12) ≈\approx 2.1734507622.173450762
L(72)L(\frac{7}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad5 1 1
13 1−169iT 1 - 169iT
good2 1+5.83iT−32T2 1 + 5.83iT - 32T^{2}
3 1+12.7iT−243T2 1 + 12.7iT - 243T^{2}
7 1+231.iT−1.68e4T2 1 + 231. iT - 1.68e4T^{2}
11 1+223.T+1.61e5T2 1 + 223.T + 1.61e5T^{2}
17 1+1.32e3iT−1.41e6T2 1 + 1.32e3iT - 1.41e6T^{2}
19 1−1.48e3T+2.47e6T2 1 - 1.48e3T + 2.47e6T^{2}
23 1+1.32e3iT−6.43e6T2 1 + 1.32e3iT - 6.43e6T^{2}
29 1+1.67e3T+2.05e7T2 1 + 1.67e3T + 2.05e7T^{2}
31 1−8.48e3T+2.86e7T2 1 - 8.48e3T + 2.86e7T^{2}
37 1−4.10e3iT−6.93e7T2 1 - 4.10e3iT - 6.93e7T^{2}
41 1+1.00e4T+1.15e8T2 1 + 1.00e4T + 1.15e8T^{2}
43 1−3.44e3iT−1.47e8T2 1 - 3.44e3iT - 1.47e8T^{2}
47 1+1.64e4iT−2.29e8T2 1 + 1.64e4iT - 2.29e8T^{2}
53 1−1.69e4iT−4.18e8T2 1 - 1.69e4iT - 4.18e8T^{2}
59 1+1.19e4T+7.14e8T2 1 + 1.19e4T + 7.14e8T^{2}
61 1−3.74e4T+8.44e8T2 1 - 3.74e4T + 8.44e8T^{2}
67 1+7.25e3iT−1.35e9T2 1 + 7.25e3iT - 1.35e9T^{2}
71 1−5.20e4T+1.80e9T2 1 - 5.20e4T + 1.80e9T^{2}
73 1−8.56e4iT−2.07e9T2 1 - 8.56e4iT - 2.07e9T^{2}
79 1−4.51e4T+3.07e9T2 1 - 4.51e4T + 3.07e9T^{2}
83 1−1.14e5iT−3.93e9T2 1 - 1.14e5iT - 3.93e9T^{2}
89 1+2.26e4T+5.58e9T2 1 + 2.26e4T + 5.58e9T^{2}
97 1−1.07e5iT−8.58e9T2 1 - 1.07e5iT - 8.58e9T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−10.17993917496046678647235101385, −9.763426304902826856961786413279, −8.014167087783643786835093898706, −7.12822444823799503551366044239, −6.68880427713110027456291027379, −4.77003152287796400863329551455, −3.70814463369224347448169985725, −2.52127245616668615014073243234, −1.20294377828739099221631865920, −0.60258683422826490855403637497, 1.97651444843720018961651315817, 3.21712749742906457199113994717, 4.87355179432097574293405335919, 5.54697395640492290602952573348, 6.37309593066170997158860837479, 7.69408060288041963285126579369, 8.485965986655146697927157017750, 9.361469959881390226977711409513, 10.32587630083256019287308998931, 11.39770328462428093684051164946

Graph of the ZZ-function along the critical line